When we analyze the dynamical state of the compressible fluid such as shock waves are arising by the conventional numerical analysis system, we need to calculate the extreme changes of physical quantities (e.g. density, pressure, velocity) if the Mach number is relatively large. The calculation grid (or mesh) of very large number of cells is required in order to calculate very accurately the part showing extreme changes of the physical quantities. But it is not economical to construct a uniform fine grid all over the whole area for analysis because of spending much memory. If the special grid suitable for the problem to be solved is made to save memory, such analysis system is not good for general use. Wide purpose and memory efficiency are important points to build a numerical analysis system.
For all-purpose numerical analysis system, the grid must be easily adaptable to various shapes in the range from simple shape to complex shape of the objects to be analyzed. Triangle has an advantage of freedom to adapt the analyzing grid to the complex-shaped object in the two-dimensional space. Tetrahedral cell and hexahedral cell (e.g. cube) are used for the cell of analyzing grid in three-dimensional space. Tetrahedral cell has an advantage of freedom in the shape of grid cell. The grid of hexahedral cell has an advantage to enable efficient calculation of analysis.
Therefore, it is the best combination for analyzing the problem of complex-shaped object to use the grid of hexahedral cells in the analyzing area as much as possible, and to use the grid of tetrahedral cells only in the area difficult to create the grid. Triangle grid and tetrahedral grid are relatively easy to vary the size of cell in the analyzing area. Triangle grid and Tetrahedral grid enable the analyzing grid to be all-purpose because the high quality grid can be created in the analyzing area in a short time even though the shape of the analyzing object is complex.
For saving memory, the grid is created fine only in the highly varying area of physical quantities. In the static analysis, it is able to create the fine grid only in the area forecast previously to arise shock waves. But in the dynamic analysis, in order to analyze the area where the region of discontinuity of physical quantities is moving such as shock wave, it is necessary to change the fine grid area according to each time.
Therefore, Solution-Adaptive grid is used in the dynamic analysis of shock wave for efficient use of memory. The area where the physical quantities are highly varying such as shock wave area is detected according to the time proceedings, and then the fine cells are concentrated only in that area by dividing and merging cells. In this way, highly accurate analysis is accomplished with the least necessary memory for calculation.
There are p-method, r-method and h-method in the Solution-Adaptive grid. In p-method, the interpolate function in the discontinuous area (area where physical quantities are highly varying) is changed to high degree. In r-method, the cells are concentrated in the discontinuous area. In h-method, the cells are refined in the discontinuous area. And h-method is called Adaptive Mesh Refinement (AMR).
In the analysis to treat discontinuity such as in shock wave, h-method to refine cells is the most effective. In the Adaptive Mesh Refinement, highly accurate analysis is accomplished with the least necessary memory for calculation. The cells are not only refined but also recovered to former cells by deleting and merging unnecessary fine cells. This recovering process is also referred to as coarsening.
There are some examples of conventional analysis method as follows. D. J. Mavriplis disclosed “Adaptive Meshing Techniques for Viscous Flow Calculations on Mixed Element Unstructured Meshes” on “International Journal for Numerical Method in Fluids” of 2000, vol. 34, pp. 93–111. An adaptive refinement strategy based on hierarchical element subdivision is formulated and implemented for meshes containing arbitrary mixtures of tetrahedra, hexahedra, prisms and pyramids. Special attention is given to keeping memory overheads as low as possible. Inviscid flows as well as viscous flows are computed on adaptively refined tetrahedral, hexahedral, and hybrid meshes.
U.S. Pat. No. 6,512,999 disclosed an apparatus for simulating turbulence. The apparatus is for simulating physical processes such as fluid flow. Fluid flow near a boundary or wall of an object is represented by a collection of vortex sheet layers. The layers are composed of a grid or mesh of one or more geometrically shaped space filling elements. The space filling elements take on a triangular shape. An Eulerian approach is employed for the vortex sheets, where a Finite Volume Method is used on the prismatic grid formed by the vortex sheet layers. A Lagrangian approach is employed for the vortical elements found in the remainder of the flow domain. To reduce the computational time, a hairpin removal scheme is employed to reduce the number of vortex filaments.
However, in the conventional analysis method, there is a problem that the efficiency of calculation is low. It is because the shape of the intermediate cell becomes very complex to connect the tetrahedral cell and hexahedral cell. So, there is little freedom in grid generation to create the suitable shape of cell. Then the cell of complex shape cannot raise the efficiency of calculation. The aim of the present invention is that both of the high freedom of grid generation and the high efficiency of calculation should be achieved employing Hybrid Grid Adaptation Method in the numerical analysis system.
In this invention, the numerical analysis system by Finite Volume Method employing Hybrid Grid Adaptation Method is constructed as follows. The initial grid data is made by way of free combination of tetrahedral cells, hexahedral cells and pentahedral cells. A cell is divided into plural cells in order to divide a triangular face of a cell into plural triangular faces and to divide a quadrilateral face of a cell into plural quadrilateral faces. The unnecessary divided cells are recovered to the former cell by deleting daughter cells. The size of the face of the cell is changed but the face shape of the cell is unchanged when the cell is divided. Any combination of various cells does not cause complexity when divided. Therefore, the freedom of grid creation is high. And as the unnecessary fine cells are deleted and recovered to the former large cell, the efficiency of calculation also becomes high.